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A120665 a(n) = 6*a(n-1) - 9*a(n-2) + 2*a(n-3) for n>3, a(1)=0, a(2)=-1, a(3)=0, 1

%I

%S 0,-1,0,9,52,231,936,3641,13884,52407,196768,736713,2754180,10288199,

%T 38415000,143404569,535268812,1997801751,7456200336,27827523881,

%U 103854943764,387594348327,1446526643848,5398520615673,20147572596060,75191803322999,280619707804800

%N a(n) = 6*a(n-1) - 9*a(n-2) + 2*a(n-3) for n>3, a(1)=0, a(2)=-1, a(3)=0,

%C Old name was: "Sequence produced by a 3 X 3 Markov chain based on a Cartan matrix."

%C Characteristic polynomial is 2 - 9 x + 6 x^2 - x^3.

%H Colin Barker, <a href="/A120665/b120665.txt">Table of n, a(n) for n = 1..1000</a>

%H MAGMA, <a href="http://magma.maths.usyd.edu.au/magma/htmlhelp/text1021.htm">MAGMA help file</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/WeylGroup.html">Weyl Groups</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-9,2).

%F Recurrence (via the Cayley-Hamilton theorem): a(n) = 6*a(n-1) - 9*a(n-2) + 2*a(n-3) (see the 2nd Maple program). - _Emeric Deutsch_, Jul 18 2007

%F O.g.f.: -x^2*(-1+6*x) / ((2*x-1)*(x^2-4*x+1)). - _R. J. Mathar_, Dec 05 2007

%F a(n) = (-2^(2+n) + (11-6*sqrt(3))*(2+sqrt(3))^n + (2-sqrt(3))^n*(11+6*sqrt(3))) / 6. - _Colin Barker_, Feb 05 2017

%p with(linalg): M := matrix(3, 3, [2, -1, 0, -1, 2, -2, 0, -1, 2]): v[1] := matrix(3, 1, [0, 1, 2]): for n from 2 to 25 do v[n] := multiply(M, v[n-1]) end do: seq(v[n][1, 1], n = 1 .. 25); # _Emeric Deutsch_, Jul 18 2007

%p a[1]:=0: a[2]:=-1: a[3]:=0: for n from 4 to 25 do a[n]:= 6*a[n-1]-9*a[n-2]+2*a[n-3] end do: seq(a[n],n=1..25); # _Emeric Deutsch_, Jul 18 2007

%t M = {{2, -1, 0}, {-1, 2, -2}, {0, -1, 2}} ; v[1] = {0, 1, 2} ; v[n_] := v[n] = M.v[n - 1] ; a = Table[ v[n][[1]], {n, 1, 50}]

%o (PARI) concat(0, Vec(-x^2*(1 - 6*x) / ((1 - 2*x)*(1 - 4*x + x^2)) + O(x^30))) \\ _Colin Barker_, Feb 05 2017

%K sign,easy

%O 1,4

%A _Roger L. Bagula_, Aug 11 2006, corrected Jul 13 2007

%E Edited by _N. J. A. Sloane_, Jul 13 2007, Jul 21 2007

%E New name from _Joerg Arndt_, Feb 05 2017

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Last modified July 6 21:41 EDT 2020. Contains 335483 sequences. (Running on oeis4.)